Theorem un0.1 45222

Description: ⊤ is the constant true, a tautology (see df-tru 1550). Kleene's "empty conjunction" is logically equivalent to ⊤. In a virtual deduction we shall interpret ⊤ to be the empty wff or the empty collection of virtual hypotheses. ⊤ in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If 𝜃 is true given the empty collection of virtual hypotheses and another collection of virtual hypotheses, then it is true given only the other collection of virtual hypotheses. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
un0.1.1	⊢ (   ⊤   ▶   𝜑   )
un0.1.2	⊢ (   𝜓   ▶   𝜒   )
un0.1.3	⊢ (   (   ⊤   ,   𝜓   )   ▶   𝜃   )

Assertion

Ref	Expression
un0.1	⊢ (   𝜓   ▶   𝜃   )

Proof of Theorem un0.1

Step	Hyp	Ref	Expression
1		un0.1.1	. . . 4 ⊢ (   ⊤   ▶   𝜑   )
2	1	in1 45015	. . 3 ⊢ (⊤ → 𝜑)
3		un0.1.2	. . . 4 ⊢ (   𝜓   ▶   𝜒   )
4	3	in1 45015	. . 3 ⊢ (𝜓 → 𝜒)
5		un0.1.3	. . . 4 ⊢ (   (   ⊤   ,   𝜓   )   ▶   𝜃   )
6	5	dfvd2ani 45027	. . 3 ⊢ ((⊤ ∧ 𝜓) → 𝜃)
7	2, 4, 6	uun0.1 45221	. 2 ⊢ (𝜓 → 𝜃)
8	7	dfvd1ir 45017	1 ⊢ (   𝜓   ▶   𝜃   )

Syntax hints:  ⊤wtru 1548  (   wvd1 45013  (   wvhc2 45024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-vd1 45014  df-vhc2 45025

This theorem is referenced by:  sspwimpVD  45362